#7484868

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,5)(3,6,7,8)(17,18,22,32,23,20,24,27)(19,28,21,26,30,25,29,31);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19,23,30)(18,25,20,28)(21,24,29,22)(26,32,31,27);(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,25,30,28)(21,26,29,31)(22,32,24,27);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,21,23,29)(18,26,20,31)(19,22,30,24)(25,27,28,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,22,23,24)(18,27,20,32)(19,29,30,21)(25,31,28,26)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7392064);